Plateaus in the Potentials of Density-Functional Theory: Analytical Derivation and Useful Approximations

Density functional theory (DFT) is an extremely efficient and widely used method for electronic structure calculations. However, the quality of such calculations crucially depends on the quality of the approximation used for the exchange-correlation functional, for which there is no exact form. One important feature of the exact exchange-correlation potential, which common approximations usually do not capture, is the spatial steps and plateaus that occur in various scenarios, including ionization, excitation, dissociation, and charge transfer. In this paper, we derive an analytical expression for the plateau in the Kohn-Sham potential that forms around the center of the system, when the number of electrons infinitesimally surpasses an integer. The resulting formula is the first analytical expression of its kind. The derivation is performed using the orbital-free DFT framework, analyzing both the Kohn-Sham-Pauli and the Pauli potentials. Analytical results are compared to exact calculations for small atomic systems, showing close correspondence and high accuracy. Furthermore, it is shown that plateaus can be produced also when relying on approximate electron densities, even those obtained with the simplest exchange-correlation form─the local density approximation.